Optimal. Leaf size=178 \[ -\frac {a d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{2 b^4}-\frac {a d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{2 b^4}+\frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {a d \sinh (c+d x)}{2 b^3 (a+b x)}-\frac {\cosh (c+d x)}{b^2 (a+b x)}+\frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2} \]
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Rubi [A] time = 0.37, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 3297, 3303, 3298, 3301} \[ -\frac {a d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{2 b^4}+\frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^3}-\frac {a d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{2 b^4}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {a d \sinh (c+d x)}{2 b^3 (a+b x)}-\frac {\cosh (c+d x)}{b^2 (a+b x)}+\frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx &=\int \left (-\frac {a \cosh (c+d x)}{b (a+b x)^3}+\frac {\cosh (c+d x)}{b (a+b x)^2}\right ) \, dx\\ &=\frac {\int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx}{b}-\frac {a \int \frac {\cosh (c+d x)}{(a+b x)^3} \, dx}{b}\\ &=\frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2}-\frac {\cosh (c+d x)}{b^2 (a+b x)}+\frac {d \int \frac {\sinh (c+d x)}{a+b x} \, dx}{b^2}-\frac {(a d) \int \frac {\sinh (c+d x)}{(a+b x)^2} \, dx}{2 b^2}\\ &=\frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2}-\frac {\cosh (c+d x)}{b^2 (a+b x)}+\frac {a d \sinh (c+d x)}{2 b^3 (a+b x)}-\frac {\left (a d^2\right ) \int \frac {\cosh (c+d x)}{a+b x} \, dx}{2 b^3}+\frac {\left (d \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2}+\frac {\left (d \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2}\\ &=\frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2}-\frac {\cosh (c+d x)}{b^2 (a+b x)}+\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^3}+\frac {a d \sinh (c+d x)}{2 b^3 (a+b x)}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {\left (a d^2 \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{2 b^3}-\frac {\left (a d^2 \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{2 b^3}\\ &=\frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2}-\frac {\cosh (c+d x)}{b^2 (a+b x)}-\frac {a d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{2 b^4}+\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^3}+\frac {a d \sinh (c+d x)}{2 b^3 (a+b x)}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {a d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{2 b^4}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 158, normalized size = 0.89 \[ -\frac {d (a+b x)^2 \left (\text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \cosh \left (c-\frac {a d}{b}\right )-2 b \sinh \left (c-\frac {a d}{b}\right )\right )+\text {Shi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \sinh \left (c-\frac {a d}{b}\right )-2 b \cosh \left (c-\frac {a d}{b}\right )\right )\right )+b \cosh (d x) (b \cosh (c) (a+2 b x)-a d \sinh (c) (a+b x))-b \sinh (d x) (a d \cosh (c) (a+b x)-b \sinh (c) (a+2 b x))}{2 b^4 (a+b x)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 373, normalized size = 2.10 \[ -\frac {2 \, {\left (2 \, b^{3} x + a b^{2}\right )} \cosh \left (d x + c\right ) + {\left ({\left (a^{3} d^{2} - 2 \, a^{2} b d + {\left (a b^{2} d^{2} - 2 \, b^{3} d\right )} x^{2} + 2 \, {\left (a^{2} b d^{2} - 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{3} d^{2} + 2 \, a^{2} b d + {\left (a b^{2} d^{2} + 2 \, b^{3} d\right )} x^{2} + 2 \, {\left (a^{2} b d^{2} + 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (a b^{2} d x + a^{2} b d\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{3} d^{2} - 2 \, a^{2} b d + {\left (a b^{2} d^{2} - 2 \, b^{3} d\right )} x^{2} + 2 \, {\left (a^{2} b d^{2} - 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{3} d^{2} + 2 \, a^{2} b d + {\left (a b^{2} d^{2} + 2 \, b^{3} d\right )} x^{2} + 2 \, {\left (a^{2} b d^{2} + 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 529, normalized size = 2.97 \[ -\frac {a b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 2 \, a^{2} b d^{2} x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 2 \, b^{3} d x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, a^{2} b d^{2} x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 2 \, b^{3} d x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + a^{3} d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 4 \, a b^{2} d x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{3} d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 4 \, a b^{2} d x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a b^{2} d x e^{\left (d x + c\right )} + a b^{2} d x e^{\left (-d x - c\right )} - 2 \, a^{2} b d {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, a^{2} b d {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a^{2} b d e^{\left (d x + c\right )} + 2 \, b^{3} x e^{\left (d x + c\right )} + a^{2} b d e^{\left (-d x - c\right )} + 2 \, b^{3} x e^{\left (-d x - c\right )} + a b^{2} e^{\left (d x + c\right )} + a b^{2} e^{\left (-d x - c\right )}}{4 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 435, normalized size = 2.44 \[ -\frac {d^{3} {\mathrm e}^{-d x -c} a x}{4 b^{2} \left (b^{2} d^{2} x^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {d^{3} {\mathrm e}^{-d x -c} a^{2}}{4 b^{3} \left (b^{2} d^{2} x^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {d^{2} {\mathrm e}^{-d x -c} x}{2 b \left (b^{2} d^{2} x^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {d^{2} {\mathrm e}^{-d x -c} a}{4 b^{2} \left (b^{2} d^{2} x^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}+\frac {d^{2} {\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right ) a}{4 b^{4}}+\frac {d \,{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right )}{2 b^{3}}+\frac {d^{2} {\mathrm e}^{d x +c} a}{4 b^{4} \left (\frac {a d}{b}+d x \right )^{2}}+\frac {d^{2} {\mathrm e}^{d x +c} a}{4 b^{4} \left (\frac {a d}{b}+d x \right )}+\frac {d^{2} {\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right ) a}{4 b^{4}}-\frac {d \,{\mathrm e}^{d x +c}}{2 b^{3} \left (\frac {a d}{b}+d x \right )}-\frac {d \,{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right )}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b \int \frac {x e^{\left (d x + c\right )}}{b^{4} d x^{4} + 4 \, a b^{3} d x^{3} + 6 \, a^{2} b^{2} d x^{2} + 4 \, a^{3} b d x + a^{4} d}\,{d x} - b \int \frac {x}{b^{4} d x^{4} e^{\left (d x + c\right )} + 4 \, a b^{3} d x^{3} e^{\left (d x + c\right )} + 6 \, a^{2} b^{2} d x^{2} e^{\left (d x + c\right )} + 4 \, a^{3} b d x e^{\left (d x + c\right )} + a^{4} d e^{\left (d x + c\right )}}\,{d x} + \frac {x e^{\left (d x + 2 \, c\right )} - x e^{\left (-d x\right )}}{2 \, {\left (b^{3} d x^{3} e^{c} + 3 \, a b^{2} d x^{2} e^{c} + 3 \, a^{2} b d x e^{c} + a^{3} d e^{c}\right )}} - \frac {a e^{\left (-c + \frac {a d}{b}\right )} E_{4}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{2 \, {\left (b x + a\right )}^{3} b d} + \frac {a e^{\left (c - \frac {a d}{b}\right )} E_{4}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{2 \, {\left (b x + a\right )}^{3} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+b\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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